Question

A manufacturer has determined that the total cost C of operating a factory is

Answer 1

We have the next cost given function:

`C=2.5x^2+75x+25000`

The average cost function is given by:

`A(x)=(C(x))/(x)`

Replace:

`A(x)=(2.5x^2+75x+25000)/(x)`

Simplify the expression:

`\begin{gathered} A(x)=(2.5x^2)/(x)+(75x)/(x)+(25000)/(x) \\ \text{Then:} \\ A(x)=2.5x+75+(25000)/(x) \end{gathered}`

Derivate A(x)

`A^(\prime)(x)=2.5(1)+0-(25000)/(x^2)`

Set A'(x)=0

`0=2.5(1)+0-(25000)/(x^2)`

Solve for x:

`\begin{gathered} 0=2.5-(25000)/(x^2) \\ -2.5=-(25000)/(x^2) \\ x^2=(-25000)/(-2.5) \\ x^2=10000 \\ \text{Take square root of both sides:} \\ \sqrt[]{x^2}=\sqrt[]{10000} \\ \text{Hence } \\ x=100 \end{gathered}`

Therefore, the average cost per unit will be minimized at 100 units of production level.